Publications

Submitted

Jussi Behrndt, Matthias Langer, Vladimir Lotoreichik, and Jonathan Rohleder
Spectral enclosures for non-self-adjoint extensions of symmetric operators
arXiv.

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Spectral enclosures for non-self-adjoint extensions of symmetric operators (jointly with J. Behrndt, M. Langer, and J. Rohleder)

The spectral properties of non-self-adjoint extensions A_[B] of a symmetric operator in a Hilbert space are studied with the help of ordinary and quasi boundary triples and the corresponding Weyl functions. These extensions are given in terms of abstract boundary conditions involving an (in general non-symmetric) boundary operator B. In the abstract part of this paper, sufficient conditions for sectoriality and m-sectoriality as well as sufficient conditions for A_[B] to have a non-empty resolvent set are provided in terms of the parameter B and the Weyl function. Special attention is paid to Weyl functions that decay along the negative real line or inside some sector in the complex plane, and spectral enclosures for A_[B] are proved in this situation. The abstract results are applied to elliptic differential operators with local and non-local Robin boundary conditions on unbounded domains, to Schrödinger operators with δ-potentials of complex strengths supported on unbounded hypersurfaces or infinitely many points on the real line, and to quantum graphs with non-self-adjoint vertex couplings.
Pavel Exner and Vladimir Lotoreichik
Optimization of the lowest eigenvalue for leaky star graphs
arXiv.

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Optimization of the lowest eigenvalue for leaky star graphs (jointly with P. Exner)

We consider the problem of geometric optimization for the lowest eigenvalue of the two-dimensional Schrödinger operator with an attractive δ-interaction of a fixed strength the support of which is a star graph with finitely many edges of an equal length L ∈ (0,∞]. Under the constraint of fixed number of the edges and fixed length of them, we prove that the lowest eigenvalue is maximized by the fully symmetric star graph. The proof relies on the Birman-Schwinger principle, properties of the Macdonald function, and on a geometric inequality for polygons circumscribed into the unit circle.
Pavel Exner, Sylwia Kondej, and Vladimir Lotoreichik
Asymptotics of the bound state induced by δ-interaction supported on a weakly deformed plane
arXiv.

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Asymptotics of the bound state induced by δ-interaction supported on a weakly deformed plane (jointly with P. Exner and S. Kondej)

In this paper we consider the three-dimensional Schrödinger operator with a δ-interaction of strength α>0 supported on an unbounded surface parametrized by the mapping ℝ² ∋ x → (x,βf(x)), where β∈[0,∞) and f:ℝ²→ℝ³, f≢0, is a C²-smooth, compactly supported function. The surface supporting the interaction can be viewed as a local deformation of the plane. It is known that the essential spectrum of this Schrödinger operator coincides with [−¼α²,+∞). We prove that for all sufficiently small β>0 its discrete spectrum is non-empty and consists of a unique simple eigenvalue. Moreover, we obtain an asymptotic expansion of this eigenvalue in the limit β→0+. On a qualitative level this eigenvalue tends to -¼α² exponentially fast as β→0+.
Pavel Exner, Vladimir Lotoreichik, and Axel Perez-Obiol
On the bound states of magnetic Laplacians on wedges
arXiv.

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On the bound states of magnetic Laplacians on wedges (jointly with P. Exner and A. Perez-Obiol)

This note is mainly inspired by the conjecture about the existence of bound states for magnetic Neumann Laplacians on planar wedges of any aperture φ∈(0,π). So far, a proof was only obtained for apertures φ≲0.511π. The conviction in the validity of this conjecture for apertures φ≳0.511π mainly relied on numerical computations. In this note we succeed to prove the existence of bound states for any aperture φ ≲ 0.583π using a variational argument with suitably chosen test functions. Employing some more involved test functions and combining a variational argument with numerical optimisation, we extend this interval up to any aperture φ ≲ 0.595π. Moreover, we analyse the same question for closely related problems concerning magnetic Robin Laplacians on wedges and for magnetic Schrödinger operators in the plane with δ-interactions supported on broken lines.
Markus Holzmann and Vladimir Lotoreichik
Spectral analysis of photonic crystals made of thin rods
arXiv.

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Spectral analysis of photonic crystals made of thin rods (jointly with M. Holzmann)

In this paper we address the question how to design photonic crystals that have photonic band gaps around a finite number of given frequencies. In such materials electromagnetic waves with these frequencies can not propagate; this makes them interesting for a large number of applications. We focus on crystals made of periodically ordered thin rods with high contrast dielectric properties. We show that the material parameters can be chosen in such a way that transverse magnetic modes with given frequencies can not propagate in the crystal. At the same time, for any frequency belonging to a predefined range there exists a transverse electric mode that can propagate in the medium. These results are related to the spectral properties of a weighted Laplacian and of an elliptic operator of divergence type both acting in L²(ℝ²). The proofs rely on perturbation theory of linear operators, Floquet-Bloch analysis, and properties of Schroedinger operators with point interactions.
David Krejcirik and Vladimir Lotoreichik
Optimisation of the lowest Robin eigenvalue in the exterior of a compact set, II: non-convex domains and higher dimensions
arXiv.

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Optimisation of the lowest Robin eigenvalue in the exterior of a compact set, II: non-convex domains and higher dimensions (jointly with D. Krejcirik)

We consider the problem of geometric optimisation of the lowest eigenvalue of the Laplacian in the exterior of a compact set in any dimension, subject to attractive Robin boundary conditions. As an improvement upon our previous work (arXiv:1608.04896, to appear in J. Convex Anal.), we show that under either a constraint of fixed perimeter or area, the maximiser within the class of exteriors of simply connected planar sets is always the exterior of a disk, without the need of convexity assumption. Moreover, we generalise the result to disconnected compact planar sets. Namely, we prove that under a constraint of fixed average value of the perimeter over all the connected components, the maximiser within the class of disconnected compact planar sets, consisting of finitely many simply connected components, is again a disk. In higher dimensions, we prove a completely new result that the lowest point in the spectrum is maximised by the exterior of a ball among all sets exterior to bounded convex sets satisfying a constraint on the integral of a dimensional power of the mean curvature of their boundaries. Furthermore, it follows that the critical coupling at which the lowest point in the spectrum becomes a discrete eigenvalue emerging from the essential spectrum is minimised under the same constraint by the critical coupling for the exterior of a ball.
Vladimir Lotoreichik
Spectral isoperimetric inequalities for δ-interactions on open arcs and for the Robin Laplacian on planes with slits
arXiv.

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Spectral isoperimetric inequalities for δ-interactions on open arcs and for the Robin Laplacian on planes with slits

We consider the problem of geometric optimization for the lowest eigenvalue of the two-dimensional Schrödinger operator with an attractive δ-interaction supported on an open arc with two free endpoints. Under a constraint of fixed length of the arc, we prove that the maximizer is a line segment, the respective spectral isoperimetric inequality being strict. We also show that in the optimization problem for the same spectral quantity, but with the constraint of fixed endpoints, the optimizer is the line segment connecting them. Furthermore, we prove that a line segment is also the maximizer in the optimization problem for the lowest eigenvalue of the Robin Laplacian on a plane with a slit along an open arc of fixed length.

Refereed publications

Jussi Behrndt, Pavel Exner, Markus Holzmann and Vladimir Lotoreichik
On the spectral properties of Dirac operators with electrostatic δ-shell interactions
to appear in J. Math. Pures Appl. arXiv.

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On the spectral properties of Dirac operators with electrostatic δ-shell interactions (jointly with J. Behrndt, P. Exner, and M. Holzmann)

In this paper the spectral properties of Dirac operators A_η with electrostatic δ-shell interactions of constant strength η supported on compact smooth surfaces in R³ are studied. Making use of boundary triple techniques a Krein type resolvent formula and a Birman-Schwinger principle are obtained. With the help of these tools some spectral, scattering, and asymptotic properties of A_η are investigated. In particular, it turns out that the discrete spectrum of A_η inside the gap of the essential spectrum is finite, the difference of the third powers of the resolvents of A_η and the free Dirac operator A₀ is trace class, and in the nonrelativistic limit A_η converges in the norm resolvent sense to a Schrödinger operator with an electric δ-potential of strength η.
David Krejcirik and Vladimir Lotoreichik
Optimisation of the lowest Robin eigenvalue in the exterior of a compact set
to appear in J. Convex Anal. arXiv.

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Optimisation of the lowest Robin eigenvalue in the exterior of a compact set (jointly with D. Krejcirik)

We consider the problem of geometric optimisation of the lowest eigenvalue of the Laplacian in the exterior of a compact planar set, subject to attractive Robin boundary conditions. Under either a constraint of fixed perimeter or area, we show that the maximiser within the class of exteriors of convex sets is always the exterior of a disk. We also argue why the results fail without the convexity constraint and in higher dimensions.
David Krejcirik, Vladimir Lotoreichik, and Thomas Ourmières-Bonafos
Spectral transitions for Aharonov-Bohm Laplacians on conical layers
to appear in Proc. Roy. Soc. Edinburgh Sect. A. arXiv.

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Spectral transitions for Aharonov-Bohm Laplacians on conical layers (jointly with D. Krejcirik and and T. Ourmières-Bonafos)

We consider the Laplace operator in a tubular neighbourhood of a conical surface of revolution, subject to an Aharonov-Bohm magnetic field supported on the axis of symmetry and Dirichlet boundary conditions on the boundary of the domain. We show that there exists a critical total magnetic flux depending on the aperture of the conical surface for which the system undergoes an abrupt spectral transition from infinitely many eigenvalues below the essential spectrum to an empty discrete spectrum. For the critical flux we establish a Hardy-type inequality. In the regime with infinite discrete spectrum we obtain sharp spectral asymptotics with refined estimate of the remainder and investigate the dependence of the eigenvalues on the aperture of the surface and the flux of the magnetic field.
Jiri Lipovsky and Vladimir Lotoreichik
Asymptotics of resonances induced by point interactions
to appear in Acta Phys. Pol. A. arXiv.
Jussi Behrndt, Pavel Exner, Markus Holzmann and Vladimir Lotoreichik
Approximation of Schrödinger operators with δ-interactions supported on hypersurfaces
Math. Nachr. 290 (2017), 1215–1248. arXiv.
Jussi Behrndt, Rupert L. Frank, Christian Kühn, Vladimir Lotoreichik and Jonathan Rohleder
Spectral theory for Schrödinger operators with δ-interactions supported on curves in ℝ³
Ann. Henri Poincaré 18 (2017), 1305–1347. arXiv.
Jussi Behrndt, Matthias Langer, and Vladimir Lotoreichik
Trace formulae for Schrödinger operators with singular interactions
The Pavel Exner Anniversary Volume, EMS (2017), 129–152. arXiv.
Jussi Behrndt, Matthias Langer, Vladimir Lotoreichik, and Jonathan Rohleder
Quasi boundary triples and semibounded self-adjoint extensions
Proc. Roy. Soc. Edinburgh Sect. A. 147 (2017), 895–916. arXiv.
Vladimir Lotoreichik and Jonathan Rohleder
Eigenvalue inequalities for the Laplacian with mixed boundary conditions
J. Differential Equations 263 (2017), 491–508. arXiv.

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Eigenvalue inequalities for the Laplacian with mixed boundary conditions (jointly with J. Rohleder)

Inequalities for the eigenvalues of the (negative) Laplacian subject to mixed boundary conditions on polyhedral and more general bounded domains are established. The eigenvalues subject to a Dirichlet boundary condition on a part of the boundary and a Neumann boundary condition on the remainder of the boundary are estimated in terms of either Dirichlet or Neumann eigenvalues. The results complement several classical inequalities between Dirichlet and Neumann eigenvalues due to Pólya, Payne, Levine and Weinberger, Friedlander, and others.
Pavel Exner and Vladimir Lotoreichik
A spectral isoperimetric inequality for cones
Lett. Math. Phys. 107 (2017), 717–732. arXiv.

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A spectral isoperimetric inequality for cones (jointly with P. Exner)

In this note we investigate three-dimensional Schrödinger operators with δ-interactions supported on C²-smooth cones, both finite and infinite. Our main results concern a Faber-Krahn-type inequality for the principal eigenvalue of these operators. The proofs rely on the Birman-Schwinger principle and on the fact that circles are unique minimisers for a class of energy functionals.
Pavel Exner, Vladimir Lotoreichik, and Miloš Tater
Spectral and resonance properties of the Smilansky Hamiltonian
Phys. Lett. A. 381 (2017), 756–761. arXiv.

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Spectral and resonance properties of the Smilansky Hamiltonian (jointly with P. Exner and M. Tater)

We analyze the Hamiltonian proposed by Smilansky to describe irreversible dynamics in quantum graphs and studied further by Solomyak and others. We derive a weak-coupling asymptotics of the ground state and add new insights by finding the discrete spectrum numerically. Furthermore, we show that the model has a rich resonance structure.
Vladimir Lotoreichik and Petr Siegl
Spectra of definite type in waveguide models
Proc. Amer. Math. Soc. 145 (2017), 1231–1246. arXiv.
Pavel Exner, Vladimir Lotoreichik, and Miloš Tater
On resonances and bound states of Smilansky Hamiltonian
Nanosystems: Physics, Chemistry, Mathematics 7 (2016), 789–802. arXiv.
Michal Jex and Vladimir Lotoreichik
On absence of bound states for weakly attractive δ'-interactions supported on non-closed curves in ℝ²
J. Math. Phys. 57 (2016), 022101. arXiv.
Vladimir Lotoreichik and Thomas Ourmières-Bonafos
On the bound states of Schrödinger operators with δ-interactions on conical surfaces
Comm. Partial Differential Equations 41 (2016), 999–1028. arXiv.
Jussi Behrndt, Gerd Grubb, Matthias Langer, and Vladimir Lotoreichik
Spectral asymptotics for resolvent differences of elliptic operators with δ and δ'-interactions on hypersurfaces
J. Spectr. Theory. 5 (2015), 697–729. arXiv.
Vladimir Lotoreichik, Hagen Neidhardt, and Igor Yu. Popov
Point contacts and boundary triples
Mathematical Results in Quantum Mechanics, Proceedings of the QMath12 Conference, P. Exner, W. König, and H. Neidhardt (eds), World Scientific, Singapore, 2015, pp. 283--293. arXiv.
Vladimir Lotoreichik and Jonathan Rohlelder
An eigenvalue inequality for Schrödinger operators with δ and δ′-interactions supported on hypersurfaces
Oper. Theory Adv. Appl. 247 (2015), 173–184. arXiv.
Jussi Behrndt, Pavel Exner, and Vladimir Lotoreichik
Schrödinger operators with δ-interactions supported on conical surfaces
J. Phys. A: Math. Theor. 47 (2014), 355202 (16pp). arXiv. (Open Access).
Jussi Behrndt, Pavel Exner, and Vladimir Lotoreichik
Schrödinger operators with δ and δ′-interactions on Lipschitz surfaces and chromatic numbers of associated partitions
Rev. Math. Phys. 26 (2014), 1450015 (43pp). arXiv.
Vladimir Lotoreichik
Lower bounds on the norms of extension operators for Lipschitz domains
Operators and Matrices 8 (2014), 573–592. arXiv.
Sylwia Kondej and Vladimir Lotoreichik
Weakly coupled bound state of 2-D Schrödinger operator with potential-measure
J. Math. Anal. Appl. 420 (2014), 1416–1438. arXiv. (Open Access).
Vladimir Lotoreichik and Sergey Simonov
Spectral analysis of the half-line Kronig-Penney model with Wigner-von Neumann perturbations
Rep. Math. Phys. 74 (2014), 45–72. arXiv.
Jussi Behrndt, Matthias Langer, and Vladimir Lotoreichik
Trace formulae and singular values of resolvent power differences of self-adjoint elliptic operators
J. London. Math. Soc. (2) 88 (2013), 319–337. arXiv.
Jussi Behrndt, Matthias Langer, and Vladimir Lotoreichik
Spectral estimates for resolvent differences of self-adjoint elliptic operators
Integral Equations and Operator Theory 77 (2013), 1–37. arXiv.
Jussi Behrndt, Matthias Langer, and Vladimir Lotoreichik
Schrödinger operators with δ and δ′-potentials supported on hypersurfaces
Ann. Henri Poincaré 14 (2013), 385–423. arXiv.
Vladimir Lotoreichik and Jonathan Rohleder
Schatten-von Neumann estimates for resolvent differences of Robin Laplacians on a half-space
Oper. Theory Adv. Appl. 221 (2012), 471–486. arXiv.
Vladimir Lotoreichik
Singular continuous spectrum of half-line Schrödinger operators with point interactions on a sparse set
Opuscula Math. 31 (2011), 615–628. (Open Access).
Jussi Behrndt, Matthias Langer, Igor Lobanov, Vladimir Lotoreichik and Igor Yu. Popov
A remark on Schatten-von Neumann properties of resolvent differences of generalized Robin Laplacians on bounded domains
J. Math. Anal. Appl. 371 (2010), 750–758. arXiv.
Igor Lobanov, Vladimir Lotoreichik, and Igor Yu. Popov
Lower bound on the spectrum of the two-dimensional Schrödinger operator with a delta-perturbation on a curve
Theor. Math. Phys. 162 (2010), 332–340.

Theses

PhD Thesis. Singular values and trace formulae for resolvent power differences of self-adjoint elliptic operators, adviser Prof. Dr. Jussi Behrndt, Graz University of Technology, 2012.
Master Thesis (in Russian). Point perturbations of Schrödinger operators on Riemannian manifolds and fractal sets, adviser Prof. Dr. Igor Yu. Popov, ITMO University, 2008.

Other publications

Jussi Behrndt, Matthias Langer, and Vladimir Lotoreichik,
Boundary triples for Schrödinger operators with singular interactions on hypersurfaces,
Nanosystems: Phys. Chem. Math. 7 (2016), 290–302.
Jussi Behrndt, Markus Holzmann, and Vladimir Lotoreichik
Convergence of 2D-Schrödinger operators with local scaled short-range interactions to a Hamiltonian with infinitely many delta-point interactions
Proc. Appl. Math. Mech. 14 (2014), 1005-1006. (Open Access).
Vladimir Lotoreichik
Note on 2D Schrödinger operators with δ-interactions on angles and crossing lines
Nanosystems: Phys. Chem. Math. 4 (2013), 1–7. arXiv. (Open Access).
Jussi Behrndt, Pavel Exner, and Vladimir Lotoreichik
Essential spectrum of Schrödinger operators with δ-interactions on the union of compact Lipschitz hypersurfaces
Proc. Appl. Math. Mech. (2013), 523–524. (Open Access).

Dr. Vladimir Lotoreichik

Department of Theoretical Physics

Nuclear Physics Institute

Academy of Sciences of the Czech Republic

Submitted

Refereed publications

Theses

Other publications

Submitted

Spectral enclosures for non-self-adjoint extensions of symmetric operators (jointly with J. Behrndt, M. Langer, and J. Rohleder)

Optimization of the lowest eigenvalue for leaky star graphs (jointly with P. Exner)

Asymptotics of the bound state induced by δ-interaction supported on a weakly deformed plane (jointly with P. Exner and S. Kondej)

On the bound states of magnetic Laplacians on wedges (jointly with P. Exner and A. Perez-Obiol)

Spectral analysis of photonic crystals made of thin rods (jointly with M. Holzmann)

Optimisation of the lowest Robin eigenvalue in the exterior of a compact set, II: non-convex domains and higher dimensions (jointly with D. Krejcirik)

Spectral isoperimetric inequalities for δ-interactions on open arcs and for the Robin Laplacian on planes with slits

Refereed publications

On the spectral properties of Dirac operators with electrostatic δ-shell interactions (jointly with J. Behrndt, P. Exner, and M. Holzmann)

Optimisation of the lowest Robin eigenvalue in the exterior of a compact set (jointly with D. Krejcirik)

Spectral transitions for Aharonov-Bohm Laplacians on conical layers (jointly with D. Krejcirik and and T. Ourmières-Bonafos)

Eigenvalue inequalities for the Laplacian with mixed boundary conditions (jointly with J. Rohleder)

A spectral isoperimetric inequality for cones (jointly with P. Exner)

Spectral and resonance properties of the Smilansky Hamiltonian (jointly with P. Exner and M. Tater)

Theses

Other publications